This invention relates to digital signal processing and in particular to an apparatus and method for approximating the magnitude of a complex number given its real and imaginary components.
Determining the magnitude of a complex number such as X+jY, which is a common calculation performed in radar signal processing applications, has been accomplished by at least three methods: the CORDIC computing technique, single interval linear approximation, and multiple interval piece-wise linear approximation.
The term CORDIC was originally defined as standing for COordinate Rotation DIgital Computer by Jack E. Volder in the September 1959 issue of IRE transactions on Electronic Computers in an article titled, "The CORDIC Trigonometric Computing Technique". The CORDIC computing technique was developed especially for use in a real-time digital computer where the majority of the computation involved the discontinuous, programmed solution of the trigonometric relationships of navigation equations and a high solution rate for the trigonometric relationships of coordinate transformation. The basic computing technique in CORDIC is a step-by-step sequence of pseudo rotations which result in an over-all rotation through a given angle or result in a final angular argument of zero. The CORDIC computing technique and variations thereof have been applied in many signal processing applications to find an approximate magnitude of a complex number by means of a process of successive rotations of its representative vector towards one of the real or imaginary component axes. Some drawbacks of CORDIC for magnitude approximation are: complexity of implementation, "growth" or multiplication by a factor related to the number of iterations, and time required for enough iterations to obtain acceptable accuracy.
A second method, widely used in radar signal processing, is single-interval linear approximation. Here, the real and imaginary components, X and Y are compared and the greater one is multiplied by a factor K.sub.G and added to a factor K.sub.L multiplied by the lesser one. When no actual multiplier is available, K.sub.G is unity and K.sub.L is chosen as 1/4, 3/8, or 1/2 to allow implementation with adders. Drawbacks include a need for comparison and multiplexing between X and Y, and poor accuracy of about 3% RMS error (over all angles) and greater than 10% peak error
A third method as shown in a patent to William M. Vojir et al, U.S. Pat. No. 4,599,701 describes a complex magnitude computing circuit that implements piecewise linear approximations to achieve high precision without the need for iteration and its processing burden. It is a generalization of the second method to multiple intervals wherein the greater (multiplied by a constant) and lesser components are further compared to determine which of two or more sets of constants K.sub.G and K.sub.L should be used in the multiplications. The increased accuracy, compared with the second method, is at the expense of greater complexity and time delay of the additional comparisons.
All three of these methods effectively require determination of the phase angle of the input vector to varying degrees of precision. Mechanization, even to simply determine the octant of the angle, requires a "pass the maximum to one destination and the minimum to another" function with considerable complexity of the requisite comparison and multiplexing hardware which must handle all bits of each component.